One needs to exercise a slight degree of restraint in adapting the
general theory of PCTPs developed for simplicial sets to arbitrary
categories \({C}\)  — not all of the theory generalizes readily.
Some arguments (most notably Lemma 18.6
of [1]}) involve constructions of pseudo-cross sections
which ultimately rely on the axiom of choice. That said, all of the
theory from [1]} that we shall need goes through.
